What is the integral of (x^3-4x+3)/(x+5) ?

The integral of (x^3-4x+3)/(x+5) is x^3+15x^2+146x+1/3 (x+5)^3-102ln|x+5|-(x+5)^3-15/2 (x+5)^2+378+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3-4x+3)/(x+5)

\int \frac{x ^{3} - 4 x + 3}{x + 5} d x

x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378 + C

Solution steps

\int \frac{x ^{3} - 4 x + 3}{x + 5} d x

Apply u-substitution

: \int \frac{- 4 u + ( u - 5 ) ^{3} + 23}{u} d u

= \int \frac{- 4 u + ( u - 5 ) ^{3} + 23}{u} d u

Apply Integration By Parts

: - 4 u + (u - 5)^{3} + 23 - \int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u

= - 4 u + (u - 5)^{3} + 23 - \int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u

\int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u = u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣

= - 4 u + (u - 5)^{3} + 23 - (u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣)

Substitute back

u = x + 5

= - 4 (x + 5) + (x + 5 - 5)^{3} + 23 - ((x + 5)^{3} - \frac{( x + 5 ) ^{3}}{3} - 15 (x + 5)^{2} + \frac{15 ( x + 5 ) ^{2}}{2} + 102 ln ∣ x + 5 ∣)

Simplify

- 4 (x + 5) + (x + 5 - 5)^{3} + 23 - ((x + 5)^{3} - \frac{( x + 5 ) ^{3}}{3} - 15 (x + 5)^{2} + \frac{15 ( x + 5 ) ^{2}}{2} + 102 ln ∣ x + 5 ∣) : x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378

= x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378

Add a constant to the solution

= x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378 + C

\int \frac{c}{r ^{2}} d r

\int r r + 3 d r

\int \frac{x ^{2} + 2 x + 1}{2 x ^{3} + 3 x ^{2} - 2 x} d x

\int \frac{3 sin ( 2 x )}{cos ( 2 x ) + 2} d x

\int \frac{2 x - 3}{x} d x

What is the integral of (x^3-4x+3)/(x+5) ?
The integral of (x^3-4x+3)/(x+5) is x^3+15x^2+146x+1/3 (x+5)^3-102ln|x+5|-(x+5)^3-15/2 (x+5)^2+378+C